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334 XI ELEMENTARY SPECTRAL THEORY
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For compact self-adjoint operators in a Hilbert space E, (11.5.7) yields
a formula for the solutions of the equation u(x) - Ax = y in E: |
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(1 1 .5.1 1 ) Let y = J]yk + ^jyk+yobe the canonical decomposition ofy in E.
k k
Then:
(a) Ifl is not in sp(w), the unique solution x of the equation u(x) - Ix = y
is given by its canonical decomposition |
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(11.5.11.1) x«£ ri + y^-yo.
k t*k A k vk A A
(b) If k is one of the eigenvalues ftk (resp. vk), then, in order that the
equation u(x) — Ix = y have a solution, it is necessary and sufficient that j;£ = 0 (resp. yk = 0). The solutions are then given by formula (11.5.11.1) in which the term corresponding to fik (resp. vk) is replaced by an arbitrary element of E(nk) (resp. E(vk)).
(c) In order that the equation u(x) = y have a solution, it is necessary and
sufficient that y0 = 0 and that the series £(1/A^)IWII2 and £(l/v£)||;y£||2 be
k k
convergent; the solutions are then given by
(11.5.11.2) x-I-ri + E-tf + Xo
k Mfc fc vfc
with x0 arbitrary in w~*(0).
Results (a) and (b) at once follow from (11 .5.7) and (1 1 .5.6), the formulae
being obtained by using the uniqueness of the canonical decomposition. The same argument proves that if there are solutions to u(x) = y, they are neces- sarily given by (11.5.11.2), hence the necessity of the conditions; and if these conditions are satisfied, then the right-hand side of (1 1 .5.1 1 .2) is an element of E (by Section 6.4) which satisfies u(x) = y. |
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PROBLEMS
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1. Let E be the vector space of all indefinitely differentiable complex valued functions
defined in the interval [0,1] of R (Section 8.12); E is made into a prehilbert space by the hermitian form |
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